** Arc length
and Curvature**:

In our treatment of differential geometry, arc length is given by:

L =** **Öå ^{3}** _{ i =1} **( y

_{i}- x

_{i })

^{2}Where (
x _{i} , y _{i}) define the end points of a curve segment. Hence, when defining the length of an arc C of a curve we can technically approximate it using a sequential series of broken lines or chords, as depicted below:

The length of such broken line can easily be determined if the end points of each chord are known. Let **ℓ**_{i} then be a series of broken lines or chords whose end points lie on C. Then if **ℓ**_{i} tends to some limit s, C is said to be rectifiable and is called the length of the arc C. Let x(t) then (a __< __ t __<__ b) be an allowable representation of an arc of a curve of class r __>__ 1. Then the arc C has length:

s = ò ^{b }_{a}_{ } **Ö**** **{å ^{3}_{ }_{i }_{=1}** **(dx^{ }_{i
}/dt) ^{2}^{ }} dt

And s is independent of the choice of allowable representation. Note that if we replace the fixed value b in the above with the variable t then s becomes a function of t. Note also that a can be replaced with any other fixed value. Thereby we obtain the integral:

^{b }

_{a}

_{ }

**Ö(**

**x'**

**·**

**x'**) dt

^{}_{o }> t then s(t) is positive and equal to the arc a(t

_{o}) b (t) of C. If t< t

_{o}then s(t) is negative and the length of a(t

_{o}) b (t) is given by - s(t). Instead of:

^{2}= å

**x**

^{3}_{ }_{i }_{=1}_{i}

_{ }

^{2 }=

**x'**

**·**

**x'**

^{2}= å

^{3}

_{ }

_{i }

_{=1}**dx**

_{i}

_{ }

^{2 }=

**d**

**x**

**· d**

**x**

**x'**

**·**

**x' =**r

^{2}+ c

^{2}

^{2}+ c

^{2}

^{2}+ c

^{2}

_{ 1}= t,

^{ }

_{ 3}= 0 (0

__<__t

__<__t

_{ 1}

*Example (3)*: If the arc length in polar coordinates can be obtained from, e.g.:

*Soln*. By integration from the polar eqn.

r(q) ^{2 } = (q - sin q) ^{2 }

= (q - sin q) (q - sin q) =

q ^{2 } - 2 q sin q +
sin ^{2 } q

And: d r(q) ** / **d q ** = **1** - **cos q

(d r(q) ** / **d q ) ^{2 } =

(1** - **cos q) (1** -** cos** ** q) = 1 - 2 cos q + cos ^{2 } q

Whence:

r(q) ^{2 } + (d r(q) ** / **d q ) ^{2 } =

q ^{2 } - 2 q sin q + sin ^{2 } q + 1 - 2 cos q + cos ^{2 } q

Leaving the integral:

ò^{ p / 2}^{ }_{0}** **Ö{( q ^{2 } - 2 q sin q +
sin ^{2 } q)
+ ( 1 - ** **2 cos q + cos ^{2 } q)} d q

=> ò^{ p / 2}^{ }_{0}** **Ö{q ^{2 } -
2 cos ** ^{ }**q -
2 q sin q
+2 } d q

(Rem: sin ^{2 }q + cos ^{2 } q = 1)

**From direct Mathcad computation:**

*And the full curve sketched*:

__:__

*Suggested Problems*_{i}by:

x_{ 1} =
t, ^{ }x_{ 2} = arcsin 1/t, x_{ 3} = (t_{ 2} – 1) ^{1/2}

Compute the length of the arc between t = 1 and t = 2

f(x) = x^{3 }/ 2 - x^{2 }/ 3

Between x= 0 and x= 2

3) Determine the arc length of a catenary with parametric representation: x(t) = (t, a cosh (t/a), 0)

4) Find the *full arc length *of the Archimedian spiral shown by changing the integral to the correct limits.

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